Theorem iota4an 5252

Description: Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
iota4an	|- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )

Proof of Theorem iota4an

Step	Hyp	Ref	Expression
1		iota4 5251	. 2 |- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) )
2		euiotaex 5248	. . . 4 |- ( E! x ( ph /\ ps ) -> ( iota x ( ph /\ ps ) ) e. _V )
3		simpl 109	. . . . 5 |- ( ( ph /\ ps ) -> ph )
4	3	sbcth 3012	. . . 4 |- ( ( iota x ( ph /\ ps ) ) e. _V -> [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) )
5	2, 4	syl 14	. . 3 |- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) )
6		sbcimg 3040	. . . 4 |- ( ( iota x ( ph /\ ps ) ) e. _V -> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) <-> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) ) )
7	2, 6	syl 14	. . 3 |- ( E! x ( ph /\ ps ) -> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) <-> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) ) )
8	5, 7	mpbid 147	. 2 |- ( E! x ( ph /\ ps ) -> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) )
9	1, 8	mpd 13	1 |- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E!weu 2054   e. wcel 2176   _Vcvv 2772   [.wsbc 2998   iotacio 5230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-sn 3639  df-pr 3640  df-uni 3851  df-iota 5232

This theorem is referenced by: (None)